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Theorem nmoofval 27957
Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1 𝑋 = (BaseSet‘𝑈)
nmoofval.2 𝑌 = (BaseSet‘𝑊)
nmoofval.3 𝐿 = (normCV𝑈)
nmoofval.4 𝑀 = (normCV𝑊)
nmoofval.6 𝑁 = (𝑈 normOpOLD 𝑊)
Assertion
Ref Expression
nmoofval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
Distinct variable groups:   𝑥,𝑡,𝑧,𝑈   𝑡,𝑊,𝑥,𝑧   𝑡,𝑋,𝑧   𝑡,𝑌,𝑥   𝑡,𝐿   𝑡,𝑀
Allowed substitution hints:   𝐿(𝑥,𝑧)   𝑀(𝑥,𝑧)   𝑁(𝑥,𝑧,𝑡)   𝑋(𝑥)   𝑌(𝑧)

Proof of Theorem nmoofval
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2 𝑁 = (𝑈 normOpOLD 𝑊)
2 fveq2 6333 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 nmoofval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2823 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54oveq2d 6812 . . . 4 (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑𝑚 𝑋))
6 fveq2 6333 . . . . . . . . . . 11 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
7 nmoofval.3 . . . . . . . . . . 11 𝐿 = (normCV𝑈)
86, 7syl6eqr 2823 . . . . . . . . . 10 (𝑢 = 𝑈 → (normCV𝑢) = 𝐿)
98fveq1d 6335 . . . . . . . . 9 (𝑢 = 𝑈 → ((normCV𝑢)‘𝑧) = (𝐿𝑧))
109breq1d 4797 . . . . . . . 8 (𝑢 = 𝑈 → (((normCV𝑢)‘𝑧) ≤ 1 ↔ (𝐿𝑧) ≤ 1))
1110anbi1d 615 . . . . . . 7 (𝑢 = 𝑈 → ((((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))))
124, 11rexeqbidv 3302 . . . . . 6 (𝑢 = 𝑈 → (∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))))
1312abbidv 2890 . . . . 5 (𝑢 = 𝑈 → {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))})
1413supeq1d 8512 . . . 4 (𝑢 = 𝑈 → sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ))
155, 14mpteq12dv 4868 . . 3 (𝑢 = 𝑈 → (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
16 fveq2 6333 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17 nmoofval.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
1816, 17syl6eqr 2823 . . . . 5 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
1918oveq1d 6811 . . . 4 (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑𝑚 𝑋) = (𝑌𝑚 𝑋))
20 fveq2 6333 . . . . . . . . . . 11 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
21 nmoofval.4 . . . . . . . . . . 11 𝑀 = (normCV𝑊)
2220, 21syl6eqr 2823 . . . . . . . . . 10 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
2322fveq1d 6335 . . . . . . . . 9 (𝑤 = 𝑊 → ((normCV𝑤)‘(𝑡𝑧)) = (𝑀‘(𝑡𝑧)))
2423eqeq2d 2781 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 = ((normCV𝑤)‘(𝑡𝑧)) ↔ 𝑥 = (𝑀‘(𝑡𝑧))))
2524anbi2d 614 . . . . . . 7 (𝑤 = 𝑊 → (((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))))
2625rexbidv 3200 . . . . . 6 (𝑤 = 𝑊 → (∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))))
2726abbidv 2890 . . . . 5 (𝑤 = 𝑊 → {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))})
2827supeq1d 8512 . . . 4 (𝑤 = 𝑊 → sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))
2919, 28mpteq12dv 4868 . . 3 (𝑤 = 𝑊 → (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
30 df-nmoo 27940 . . 3 normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
31 ovex 6827 . . . 4 (𝑌𝑚 𝑋) ∈ V
3231mptex 6633 . . 3 (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )) ∈ V
3315, 29, 30, 32ovmpt2 6947 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 normOpOLD 𝑊) = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
341, 33syl5eq 2817 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062   class class class wbr 4787  cmpt 4864  cfv 6030  (class class class)co 6796  𝑚 cmap 8013  supcsup 8506  1c1 10143  *cxr 10279   < clt 10280  cle 10281  NrmCVeccnv 27779  BaseSetcba 27781  normCVcnmcv 27785   normOpOLD cnmoo 27936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-sup 8508  df-nmoo 27940
This theorem is referenced by:  nmooval  27958  hhnmoi  29100
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